20
Metonymy and its relatives, metaphor, polysemy, synecdoche occur all over the place in mathematical writing, and sometimes cause students problems and sometimes don't, because those thought processes are basic to our understanding of everything.
Where mathematics comes from, by George Lakoff and Rafael E. Núñez. Basic Books, 2000.
Exploring the role of ...
19
I don't see this as a major issue, nor do I believe that the word "unique" is in any particular need of saving. There are a large number of terms in mathematics which correspond to vernacular English words, but which have distinct technical meaning in mathematics. For example, an "odd" number is an integer which is not a multiple of $2$,...
17
Perhaps not pointing out that the obvious steps are obvious but that the insights are insights.
I believe students don't feel bad for not seeing the "magic steps" by themselves, so pointing out that those are hard is not a problem. The opposite is what you mention: they would feel bad for not seeing the obvious. Hence, only treat the difficult as difficult.
16
Not formal research, but some decades of experience teaching both undergrad and graduate level courses, and "editing" PhD theses and such:
It appears that even many serious professional mathematicians do not understand the difference between a "definitional" iff and an "assertive" iff. This is entirely parallel to an assignment equality versus an assertive ...
13
One of the most colorful names I have heard is the Chicken Mc Nugget theorem:
for any two relatively prime positive integers $m,n$, the greatest integer that cannot be written in the form $am + bn$ for nonnegative integers $a, b$ is $mn-m-n$.
link1, link2.
From the links:
The story goes that the Chicken McNugget Theorem got its name because in McDonalds, ...
13
So German "$b$ kürzt sich weg" becomes in English "$b$ cancels out". We may also say "$b$ is eliminated".
11
In Central Mexico, the expression
\begin{equation}
x_{\pm} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\end{equation}
that solves quadratic equations of the form $ax^2 + bx + c = 0$ is called "fórmula del chicharronero" (formula of the chicharronero).
The chicharronero is the guy who sells salty snacks made of wheat (called chicharrones). Outside most schools ...
11
One thing that you might want to do early on in your course is think about the classroom norms that you wish to establish. From your post, it seems like an example of a norm in your class is that it is okay to ask a question of the form:
What do you mean by obvious (routine, etc)?
If you do not try to make it clear to students from the outset that such ...
11
The ambiguous answer is relatively correct (actually you need to know how old they are at the start of the problem). But each fission is an individual splitting, not a generation.
Perhaps this:
A certain species of bacteria splits into two cells ( a process called fission) every 20 minutes (the "generation" length), assuming proper growth ...
10
For presenting proofs, I would only use terminology such as "obviously", "clearly", "routine", etc if it would be for a course just below the class had the student had 'perfect' recollection. For instance, in a introductory commutative algebra class, if the proof should be routine for the previous regular algebra class (i.e., something involving a course ...
10
How about the shoelace formula for the area of an arbitrary
simple polygon?
(Image from Wikipedia.)
The formula computes the area from the coordinates of the vertices,
essentially by a cross product to compute (signed) areas of triangles.
10
If you continue the operations until
$$\require{cancel}\frac{x}{b}=\frac{c}{b},\qquad\left(\frac{x}{b}\right)b=\left(\frac{c}{b}\right)b,\qquad x\left(\frac{b}{b}\right)=c\left(\frac{b}{b}\right),\qquad x\left(\frac{\cancel b}{\cancel b}\right)=c\left(\frac{\cancel b}{\cancel b}\right)$$
then $x=c$, I would say that you cancelled the common factor $b$ in the ...
8
If you simplify a term by adding and subtracting something you call this a "nahrhafte Null" in German (probably translates to "nutritious null"?).
8
I often refer to the identities $(AB)^{-1} = B^{-1}A^{-1}$ or $(AB)^T = B^TA^T$ as the socks-shoes identity. I'm not sure how wide-spread this is, I certainly did not invent it and I'm pretty sure I've read at least one of these in at least one text.
8
In Russian, the Squeeze Theorem (a.k.a. The Pinching Theorem) is called "Теорема о двух милиционерах" — "Two Policemen Theorem". The idea is that if two policemen are holding a criminal between them, the bad guy is going to the same place, probably jail or precinct, where the policemen are going.
8
In general, as others have noted, if you have an equation such as
$$\frac{x}{b}=\frac{c}b$$
The step to get from there to
$$x=c$$
is typically referred to as cancelling the denominator. More generally, if you can just remove some piece of the equation, you can use the verb "cancel" both with that piece as the object and the subject. For instance:
We cancel ...
7
Some examples:
"One plus five squared" could be read as $(1+5)^2$ or $1+5^2$. This is a well-known ambiguity in natural language, for example in the following sentence
I saw a man on a hill with a telescope.
we don't know if it was a man who had the telescope, or the speaker, or perhaps the telescope was just standing on some hill.
See also this list.
...
7
The argument has been made that this is sort of a misappropriation of the terms, because the levels are meant to define levels of understanding rather than levels of detail. I'm assuming what you're aiming for is that you can say to your student(s) "I'm looking for a level 2 explanation here", and they would provide you with an answer that would ...
7
To my mind the defintion "Every x has a unique f(x)" of one-to-one is problematic because "has a unique" is neither clear English nor precise. The definition is usually stated as "$f(x) = f(y)$ implies $x = y$", which is unambiguous and avoids any potential interpretative problems related to the use of the word "unique. Alternatively, one could define $f$ to ...
6
My elementary students always wanted to know the name of the symbol shown here:
We called it the division house as did many of my colleagues, but my students wanted a mathematical name. We therefore wrote to Dr. Math at Drexel. We were told there is no name and were referred to this paper.
I subsequently held a contest (on election day) and the winner ...
6
iff and if
In my experience, students who have a solid grasp of first-order logic have absolutely no problem with the inconsistent use of "if" in definitions. The problem is that most students don't, and as I've personally observed, often confuse between "if" and "iff" precisely because of such notational issues. Therefore the ...
6
I've just remembered that "Donkey Theorem" is used to refer to triangle inequality in geometry textbooks in Iran. The name implies that even a donkey which is on one corner of a triangle chooses the straight path (rather than the broken one) to get to the other corner where there is some hay to eat.
I checked to see if it is used elsewhere and I learned ...
6
As a non-native speaker I'd tend towards particular in this context. Any 2x2 matrix would be a specific matrix, but the one used by the student was a particularly bad example.
Citing Merriam Webster:
[particular]: distinctive among other examples or cases of the same general category: notably unusual
To me, specific is "any fixed number/matrix/...", ...
6
If your students are willing to take the time, I would say you can add a lot of value to their understanding and skills by aproaching the challenge from a "socratic" point of view.
Facilitate conceptualization through "concept cards":
I have my students make a mindmap with the math concept in the center.
On one side, 2-5 examples.
On the other hand, the ...
6
I highly suggest reading the book How to Teach Mathematics by Steven G. Krantz, published by the American Mathematical Society (AMS).
https://www.amazon.com/How-Teach-Mathematics-Steven-Krantz/dp/1470425521
https://www.maa.org/press/maa-reviews/how-to-teach-mathematics
6
I'll give you the answer I got from several professors when I started teaching:
"talk slow and pause" and "write what you say and say what you write"
6
It's possible that she said, or meant to say, "into", which denoted multiplication in the late 19th and early 20th centuries. Perhaps she was taught using rather antiquated terminology.
"The sign $\times$ is called the sign of multiplication, and $a \times b$ is read thus "$a$ into $b$." " Algebra for Beginners, I. Todhunter, 1872.
"Multiplication, $\...
6
If you feel your instruction is unclear, then provide an example.
Write the following sets of numbers represented in interval notation. For example, "all real numbers from 0 to 1 exclusive" in interval notation is $(0,1)$.
a) All real numbers greater than or equal to 5 and less than 7.
b) All real numbers from 1 to 10 inclusive.
5
I find the easiest way is to just straight out say that it's not something you expect them to know offhand or immediately understand, but that as their studies advance it will become trivial. That way they understand that it's new, but nothing to be scared of. Something along the lines of:
"This may not make sense to you yet, but trust me it will soon be ...
5
It may help your students if you make explicit why it is obvious/routine/etc. Have they seen it in the previous example? Does it just depend on unpacking definitions? It is a calculation/result/etc that was covered in a prerequisite class?
I don't think there is anything wrong with pointing out which parts are routine, but saying why it's routine will help ...
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